Find all continuous function $f: \mathbb R \to \mathbb R$ satisfying $f(x+y)=2+f(x)+f(y)$ and $f(3)=5$

Question

Find all continuous function $f: \mathbb R \to \mathbb R$ for which $f(3)=5$ and for every $x,y \in \mathbb R$ it is true that $f(x+y)=2+f(x)+f(y)$.

I tried to find some dependence before $x$ and $y$ because I have for example $f(1+2)=2+f(1)+f(2)=5$ and then $f(1)+f(2)=3$. However I cannot find dependence for every $x,y$ so I don't know how I can do this task.

I also thought about creating a new function $g$ which is dependent on $f$ but I also don't have a good idea to do this.

Can you get me some tips?

Answer 1

As transformed by @Hagen von Eitzen into the Cauchy functional equation (https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation). The only continous solution to Cauchy's functional equation is $g(x)=kx$. Therefore, $f(x)+2=kx$. Now given that $f(3)=5$ so that $k=\frac{7}{3}$. Thus the required function is $f(x)=\frac{7}{3}x-2$.

Answer 2

Hint: With $g(x):=f(x)+2$, we have$$ g(x+y)=g(x)+g(y).$$